11 REGULARiZATION 297
(14.11.3) Take G = R, and p to be Lebesgue measure. Put
,,,,1.3,) ,*)
Let an = )_10n(x) dx, and/n = a~lgn . The sequence (/„) satisfies the condi-
tions of (14.11.1). For 1 - x2 ^ 1 - \x\ for 1 g x g 1, hence
and therefore /„(» <£ (w + 1)(1 - x2)n for all x e [-1, 1], which proves that
fn(x) -» 0 uniformly on every compact interval not containing 0. Let ^ be a
measure on R with support contained in [— £, £]. Then we have
fl/
/„)(*) = a;1
J-l
l/2
*/„)(
l/2
and if x e [~i, i] this gives
fl/2
GI * /„)(*) = ^ M (i - (x - jO2
J-l/2
showing that the function ju */„ is equal to a, polynomial function on [—i, i].
In particular, if ^ = h - /?, where /lisa continuous function with support con-
tained in [ — i, i], we obtain from (14.11.1(i)) the theorem of Weierstrass on
uniform approximation of continuous functions by polynomials on a compact
interval (7.4.1).
PROBLEMS
1. If a locally compact group G is such that the algebra Jf(G) is commutative with
respect to convolution, show that G is commutative. (Show by regularization that the
algebra of measures with compact support is commutative.)
2. Let G be a locally compact group, j8 a left Haar measure on G. Show that the algebra
L!(G,^) Jias a unit element if and only if G is discrete. (Suppose that G is not discrete,
and let/o 6 ^KG,/?); then there exists a compact neighborhood V of e such that
I 1/oWI dB(x) < 1. Show that, if U is a compact symmetric neighborhood of e such
Jv
that U2 c V, then K^u */0)W | < 1 for almost all x e U, and hence that /0 cannot be
the unit element of L*(G, /?).) uniformly to g on every compact subset of G. If g is uniformly